Sonja Nikolić

On the Harary index for the characterization of chemical graphs

A novel topological index for the characterization of chemical graphs, derived from the reciprocal distance matrix and named the Harary index in honor of Professor Frank Harary, has been introduced. The Harary index is not a unique molecular descriptor; the smallest pair of the alkane trees with identical Harary indices has been detected in the octane family. The use of the Harary index in the quantitative structure-property relationships is exemplified in modeling physical properties of the C2-C9 alkanes. In this application, the performance of the Harary index is comparable to the performance of the Wiener number.

The distance matrix in chemistry

The graph-theoretical (topological) distance matrix and the geometric (topographic) distance matrix and their invariants (polynomials, spectra, determinants and Wiener numbers) are presented. Methods of computing these quantities are discussed. The uses of the distance matrix in both forms and the related invariants in chemistry are surveyed. Special attention is paid to the 2D and 3D Wiener numbers, defined respectively as one half of the sum of entries in the topological distance matrix and the topographic distance matrix. These numbers appear to be very valuable molecular descriptors in the structure property correlations.

On the three-dimensional Wiener number

A novel approach to the Wiener number is described. Iï is based on the distance matrix in which topographic (geometric) distances rather than topological (graph-theoretical) distances are the input entries. The Wiener number defined in this novel way is thus the representative of 3D (topographic) molecular descriptors. This novel Wiener number is tested in quantitative structure-property relationships (QSPR) with enthalpy functions of the lower alkanes. Its performance is compared to that of the traditional 2D Wiener number. The statistical analysis favours the QSPR models with the 3D Wiener numbers over the related QSPR models with the 2D Wiener numbers. Among the considered models with the 3D Wiener numbers, the best agreement with experimental enthalpy functions is obtained with the logarithmic QSPR model.

THE VERTEX-CONNECTIVITY INDEX REVISITED

We report a search for optimum molecular descriptors based on the connectivity index. A suggestion made by several authors that the exponent -0.5 used in the standard formula for computing the connectivity index may not be the optimum for modeling some molecular properties was reexamined. We considered several molecular properties and found that in most cases the optimum value of the exponent is indeed different from -0.5. We suggest that a modified version of the (valence) vertex-connectivity index should be routinely employed in the structure-property modeling instead of the standard version of the index.

The Laplacian matrix in chemistry

The Laplacian matrix, its spectrum, and its polynomial are discussed. An algorithm for computing the number of spanning trees of a polycyclic graph, based on the corresponding Laplacian spectrum, is outlined. Also, a technique using the Le Verrier-Faddeev-Frame method for computing the Laplacian polynomial of a graph is detailed. In addition, it is shown that the Wiener index of an alkane tree can be given in terms of its Laplacian spectrum. Two Mohar indices, one based on the Laplacian spectrum of a molecular graph G and the other based on the Laplacian x2 eigenvalue of G, have been tested in the structure-property relationships for octanes.

On reformulated Zagreb indices

Zagreb indices were reformulated in terms of the edge-degrees instead of the vertex-degrees as the original Zagreb indices. Three types of Zagreb indices were considered: original, modified and variable Zagreb indices. It is found that the optimum exponent of the variable reformulated Zagreb M2 index (v=−1/2) is identical with the exponent of the vertex-connectivity index, which is the most used topological index in QSPR and QSAR. The close relationship between the graph and its line graph is used to relate the original and reformulated indices.

The Detour Matrix in Chemistry

The detour matrix of a (chemical) graph is defined. The detour matrix is also defined for weighted graphs. A novel method of computing the detour matrix is introduced. Some properties of the detour matrix and the distance matrix are compared. The invariants of the detour matrix (detour polynomial, detour spectrum, and detour index) are discussed, and several methods for computing these quantities are presented. The use of the detour index is analyzed and compared to the application of the Wiener number in the structure−boiling point modeling.

The conjugated-circuit model: application to benzenoid hydrocarbons

Abstract The conjugated-circuit model is presented and its application to benzenoid hydrocarbons is described. The set of conjugated circuits is truncated at those around three hexagons and these are used for computing the resonance energies (RE) of benzenoids. The concept of benzenoidicity defined in terms of conjugated circuits is also presented. Clars concept of fully benzenoid hexagonal structures is extended to fully naphthalenoid structures and is further generalized to the concept of fully arenoid hexagonal structures. A comparison is given between the conjugated-circuit model and several other theoretical models for computing the REs of benzenoid hydrocarbons.

Wiener index revisited

Abstract We introduce a modification of the Wiener index. This modified Wiener index has a structural interpretation in terms of the greater weights of outer than inner bonds in a saturated hydrocarbon. Both the Wiener index and its modification together with the molecular polarity index lead to comparable structure-property models.

The conjugated-circuit model

Several annelated [n]annulenes are examined from a graph-theoretical point of view. It is shown how a new interpretation of the conjugated circuit model can be used in order to study the geometries of such compounds. The work illustrates which energetical factors determine whether a structure with alternating bond lengths, or with reduced symmetry is more stable than one with full double bond delocalization or full symmetry.